Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes

Technical Summary: Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes

Problem Statement
The paper investigates the relative form of $\mathbb{A}^1$-contractibility for smooth morphisms $f: X \to S$ within the unstable $\mathbb{A}^1$-homotopy category $\mathcal{H}(S)$. While $\mathbb{A}^1$-contractible varieties over fields have been extensively studied—revealing a dichotomy between rigidity in low dimensions and the existence of exotic non-isomorphic varieties in higher dimensions—the behavior of these properties over general base schemes $S$ (particularly those of finite Krull dimension) remains largely unexplored. The central questions addressed are:

1. Is $\mathbb{A}^1$-contractibility a fiberwise property for smooth morphisms?
2. What geometric structures characterize $\mathbb{A}^1$-contractible morphisms, specifically in the context of $\mathbb{A}^n$-fiber spaces?
3. To what extent do rigidity results known over fields (e.g., the uniqueness of $\mathbb{A}^1$ and $\mathbb{A}^2$) extend to relative settings, and how does the characteristic of the base scheme influence these results?

Methodology
The authors employ a combination of unstable motivic homotopy theory, base change techniques, and classical affine algebraic geometry.

• Homotopy-Theoretic Tools: The study relies on the functorial properties of the unstable $\mathbb{A}^1$-homotopy category, specifically base change along smooth morphisms and localization with respect to Nisnevich hypercovers. A key technique involves a gluing argument along a stratification of the base scheme, reducing global questions to local schemes. This approach refines analogous results from stable motivic homotopy theory to the unstable setting.
• Algebraic Geometry: The structural analysis of $\mathbb{A}^n$-fiber spaces builds upon Asanuma's work on quasi-polynomial algebras and the classification of affine fibrations. The authors utilize the Jouanolou–Thomason trick (replacing schemes with affine torsors under vector bundles) and results on vector bundle torsors.
• Dimensional Analysis: The paper distinguishes sharply between low relative dimensions ($d \leq 2$) and higher dimensions ($d \geq 3$). In low dimensions, the authors integrate fiberwise criteria with classical theorems by Kambayashi, Miyanishi–Sugie, Russell, and Sathaye. In positive characteristic, they construct counterexamples using purely inseparable morphisms and non-trivial forms of affine spaces.
• Geometry at Infinity: For the classification of surfaces, the paper utilizes techniques from [DDØ22] relating $\mathbb{A}^1$-contractibility to the stable $\mathbb{A}^1$-homotopy type of punctured tubular neighborhoods of the boundary in a compactification.

Key Contributions and Results

1. Fiberwise Characterization:
   The primary theoretical result establishes that for a smooth morphism $f: X \to S$ over a qcqs base scheme $S$ of finite Krull dimension, $f$ is $\mathbb{A}^1$-contractible if and only if every geometric fiber $X_s$ is $\mathbb{A}^1$-contractible over the residue field $\kappa(s)$. This confirms that $\mathbb{A}^1$-contractibility is a fiberwise property, implying that the homotopy category $\mathcal{H}(S)$ does not detect variations in the isomorphism types of fibers within a smooth family of $\mathbb{A}^1$-contractible varieties.

2. Structure of $\mathbb{A}^n$-Fiber Spaces:
   The authors provide a geometric description of $\mathbb{A}^n$-fiber spaces (smooth morphisms with fibers isomorphic to affine space). Under mild hypotheses (e.g., $f$ affine or $S$ regular), such spaces admit a Zariski local factorization as a tower of torsors under vector bundles. This result, building on Asanuma's work, explains the ubiquity of $\mathbb{A}^1$-contractible morphisms arising from affine fibrations.

3. Low-Dimensional Rigidity:

   • Relative Dimension 1: Over a normal base scheme $S$, a smooth morphism of relative dimension 1 is $\mathbb{A}^1$-contractible if and only if it is a Zariski locally trivial $\mathbb{A}^1$-bundle. Furthermore, if the relative canonical sheaf is trivial and a section exists, the morphism is isomorphic to the trivial bundle $\mathbb{A}^1_S$.
   • Relative Dimension 2 (Characteristic Zero): Over bases with characteristic zero residue fields, $\mathbb{A}^1$-contractibility forces the structure of an $\mathbb{A}^2$-fiber space. Over Dedekind schemes, such morphisms are Zariski locally trivial if they are affine. Consequently, smooth affine $\mathbb{A}^1$-contractible surfaces over such bases are isomorphic to $\mathbb{A}^2_S$.
   • Rigidity vs. Cancellation: The paper establishes relative versions of the Zariski Cancellation Problem for curves and surfaces in characteristic zero, showing that $X \times_S \mathbb{A}^n_S \cong \mathbb{A}^{n+d}_S$ implies $X \cong \mathbb{A}^d_S$ under appropriate conditions.

4. Pathologies in Positive and Mixed Characteristic:
   The paper demonstrates that the rigidity observed in characteristic zero fails in positive and mixed characteristic.

   • Non-Triviality: There exist $\mathbb{A}^2$-fiber spaces over discrete valuation rings with positive characteristic residue fields that are not Zariski locally trivial.
   • Cancellation Failure: Counterexamples are exhibited where $X \times_S \mathbb{A}^1_S \cong \mathbb{A}^3_S$ but $X \not\cong \mathbb{A}^2_S$.
   • Exotic Surfaces: The authors construct smooth, $\mathbb{A}^1$-connected affine surfaces in positive characteristic that are not isomorphic to $\mathbb{A}^2$ and do not admit dominant separable morphisms from $\mathbb{A}^1$. The question of whether such surfaces can be $\mathbb{A}^1$-contractible remains open.

Significance and Claims
The paper claims to develop a structural theory for $\mathbb{A}^1$-contractible morphisms over general bases, refining the understanding of how motivic homotopy types reflect algebraic geometry in families.

• It establishes that the "fiberwise" principle, familiar in topology, holds for $\mathbb{A}^1$-contractibility in the unstable setting, provided the base has finite Krull dimension.
• It highlights a sharp dichotomy: while dimensions $d \geq 3$ admit abundant non-trivial families of $\mathbb{A}^1$-contractible varieties, dimensions $d \leq 2$ exhibit strong rigidity in characteristic zero, mirroring classical results in affine algebraic geometry.
• The work underscores the limitations of extending characteristic zero results to positive characteristic, specifically regarding the classification of surfaces and the validity of cancellation properties.
• The authors formulate open problems concerning the existence of "exotic" $\mathbb{A}^1$-contractible surfaces in positive characteristic, suggesting that the classification of such objects is incomplete and potentially richer than in the characteristic zero case.

The paper does not claim to resolve the existence of exotic $\mathbb{A}^1$-contractible surfaces in positive characteristic but rather identifies the specific obstructions (separability issues) and constructs candidates that challenge current classification expectations.